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Question: Answered & Verified by Expert
$A=\left[\begin{array}{ll}3 & 1 \\ 0 & 4\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 1 \\ 0 & 2\end{array}\right]$, then which of the following is/are correct?
I. $A B$ is defined
II. $B A$ is defined
III. $A B=B A$
Select the correct answer using the codes given below.
MathematicsMatricesNDANDA 2011 (Phase 1)
Options:
  • A Only I
  • B Only II
  • C Both I and II
  • D I, II and III
Solution:
2245 Upvotes Verified Answer
The correct answer is: I, II and III
Let $\mathrm{A}=\left[\begin{array}{ll}3 & 1 \\ 0 & 4\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}1 & 1 \\ 0 & 2\end{array}\right]$
Now, $\mathrm{AB}=\left[\begin{array}{ll}3 & 1 \\ 0 & 4\end{array}\right]\left[\begin{array}{ll}1 & 1 \\ 0 & 2\end{array}\right]=\left[\begin{array}{ll}3 & 5 \\ 0 & 8\end{array}\right]$
and $\mathrm{BA}=\left[\begin{array}{ll}1 & 1 \\ 0 & 2\end{array}\right]\left[\begin{array}{ll}3 & 1 \\ 0 & 4\end{array}\right]=\left[\begin{array}{ll}3 & 5 \\ 0 & 8\end{array}\right]$
$\Rightarrow \quad \mathrm{AB}=\mathrm{BA}$
Hence, all the three statements are correct.

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